Uniqueness of measure on quotient I am trying to work through Serge Lang's book $SL_2$. On page 37, he considers the invariant measure on a quotient space. 
Let $G$ be a locally compact group with closed subgroup $K$. Let $\mu_G$ and $\mu_K$ denote respective Haar measures, where the latter is normalized to one. For a compactly supported function $f \in C_c(G)$ define
\begin{align*}
f^K(x) = \int_K f(xk) \; d\mu_K(k)
\end{align*}
It is easy to see that $f^K$ can be seen as a function on $G/K$ because $f^K(x) = f^K(xk)$ for all $k \in K$. 
Then Lang states, as a Theorem, that there is a unique invariant measure $\mu_{G/K}$ on the quotient $G/K$ such that for any $f \in C_c(G)$ we have 
\begin{align*}
\int_{G/K} f^K \; d\mu_{G/K} = \int_G f \; d\mu_G
\end{align*}
Now here is my problem: In the proof, he states that uniqueness is obvious. I must be missing some obvious fact here, because I don't see where this is coming from. I can extend every indicator function on $G/K$ to a function on $G$, but that only gives me uniqueness on bounded sets, because the functions here all have compact support. What am I missing, how is the uniqueness obvious?
Edit: I forgot to mention and it is important here that, for simplicity, he assumes unimodular groups $K$ and $G$.
 A: I think it's not really so obvious...
First, at least in your quote from Lang, there're some imprecisions. Ok, for $G$ a (locally convex, Hausdorff, probably countably-based) topological group, and $H$ a closed subgroup, for existence of a measure on $H\backslash G$ such that
$$
\int_G f \;=\; \int_{H\backslash G}\int_H f(h\dot{g})\;dh\,d\dot{g}
$$
for $f$ compactly-supported continuous on $G$, it is necessary and sufficient that the modular function of $G$, restricted to $H$, is equal to the modular function of $H$. For compact $H$ (as was probably intended in your context, by "$K$"), this condition is automatic.
Apart from the condition on modular functions, existence and uniqueness will follow from the Riesz-Kakutani-Markov theorem (that functionals on continuous compactly-supported functions are given by integrals...), once we know that the averaging map $f\to f^H$ is a surjection from $C^o_c(G)$ to $C^o_c(H\backslash G)$. And this is true, but requires a little doing.
The above (and many related basic points) is very standard, so you should be able to find many alternative sources at the points where Lang becomes impatient or too terse.
EDIT: specific references: First, yes, indeed, this issue is often imbedded in a fairly sophisticated context, and can get lost. Some of my own notes aim to be minimalist, in a good way, in dealing with some of these things. E.g., in an old note on repns of totally disconnected groups http://www.math.umn.edu/~garrett/m/v/smooth_of_td.pdf there is a proof of this. Also, in both the physical book (by Cambridge Univ Press) and my (legal!) on-line version of my book "Modern Analysis of Automorphic Forms by Example", this and other things are dealt with at a level intended to be legitimate without going overboard. E.g., chapter 5 is "integration on quotients" in http://www.math.umn.edu/~garrett/m/v/current_version.pdf
